PDF《Pure Elastic Contact Force Models》

Chapter

2 Pure Elastic Contact Force Models Abstract The most important pure elastic constitutive laws commonly utilized to model and analyze contact-impact events in the context of multibody mechanical system dynamics are presented in this chapter.

Additionally, the fundamental issues related to the generalized contact kinematics, developed under the framework of multibody system dynamics formulation, are brie?y described. In this process, the main contact parameters are determined, namely the indentation or pseudo-penetration of the potential contacting points, and the normal contact velocity. Subsequently, the linear Hooke'

s contact force model and the nonlinear Hertz'

s law are presented together with a demonstrative example of application. Some other elastic contact force models are also brie?y described. Keywords Multibody dynamics ? Contact kinematics ? Elastic contact force models ? Hooke force model ? Hertz force model 2.1 Generalized Contact Kinematics The generalized contact kinematics between two planar rigid bodies that can experience an oblique eccentric impact is ?rst described. Figure 2.1a shows two convex bodies i and j in the state of separation that are moving with absolute velocities _ ri and _ rj, respectively. The potential contact points are denoted by Pi and Pj (Machado et al. 2012). The evaluation of the contact kinematics involves the calculation of three fun- damental quantities, namely the position of the potential contact points, their Euclidian distance and their relative normal velocity (Glocker 2001;

Machado et al. 2010). In general, this information must be available in order to allow the deter- mination of the contact forces that develop during the contact-impact events (Lankarnai and Nikravesh 1990;

Gilardi and Sharf 2002;

Hippmann 2004;

Askari et al. 2014). The possible motion of each body in a multibody system can be quanti?ed in terms of the distance and relative velocity of the potential contact points. Positive values of that distance represent a separation, while negative values ? Springer International Publishing Switzerland

2016 P. Flores and H.M. Lankarani, Contact Force Models for Multibody Dynamics, Solid Mechanics and Its Applications 226, DOI 10.1007/978-3-319-30897-5_2

15 denote relative indentation or penetration of the contacting bodies. These two scenarios are illustrated in Fig. 2.1a, b, respectively. The change in sign of the normal distance indicates a transition from separation to contact, or vice versa (Flores and Ambrósio 2010). In turn, positive values of the relative normal velocity between the contact points, that is, the indentation or penetration velocity, indicate that the bodies are approaching, which corresponds to the compression phase , while negative values denote that the bodies are separating, that corresponds to the restitution phase . The vectors of interest in studying contact-impact events are represented in Fig. 2.1. The vector that connects the two potential contact points, Pi and Pj, is a gap function that can be expressed as (Nikravesh 1988) d ? rP j ? rP i ?2:1? where both rP i and rP j are described in global coordinates with respect to the inertial reference frame, that is rP k ? rk ? Ais0P i k ? i;

j ? ? ?2:2? in which ri and rj represent the global position vectors of bodies i and j, while s0P i and s0P j are the local components of the contact points with respect to local coor- dinate systems. The planar rotational transformation matrices Ak are given by (Nikravesh 1988;

Flores 2015) (a) (b) Fig. 2.1 a Two bodies in the state of separation;